Buoyancy frequency of an ideal-gas stratification ($N^2=g\left(\frac{1}{\gamma }\frac{d\ln p}{dz}-\frac{d\ln \rho }{dz}\right)$)

For uniform composition, upward coordinate $z$ and downward acceleration $g>0$, a pressure-equilibrated adiabatic parcel in a hydrostatic ideal gas has $\ddot{\zeta }=-N^2\zeta$ with $N^2=g[(d\ln p/dz)/\gamma -d\ln \rho /dz]$. A positive value gives restoring buoyancy; for a calorically perfect gas it is also $(g/c_p)ds/dz$. Heat or composition exchange requires a modified stability test.